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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Suppose that a and b are nonzero vectors that are not parallel and c is any vector in the plane determined by a and b. Give a geometric argument to show that c can be written as c = sa + tb for suitable scalars s and t. Then give an argument using components.

To determine

To give: A geometric argument to show the vector c=sa+tb.

Explanation

Given:

Non zero and non parallel two-dimensional vectors a and b and any vector c.

Definition:

Scalar multiplication:

Consider a scalar c and a vector v. The scalar multiple cv, which is a vector with a length more than |c| times of vector v in same direction of v.

Consider two-dimensional vectors a=a1,a2, b=b1,b2, and c=c1,c2. The vectors are located in xy-plane as they are two-dimensional. Sketch the vectors a, b, and c in xy-plane from origin. Name the origin as O. Extend the vector a using line A and vector b using B.

Draw the parallel lines A and B to vectors a and b from terminal point of vector c. Name the intersecting point of lines A and A as P and intersecting point of B and B as Q.

From explanation, sketch of vectors is shown in Figure 1.

From Figure 1, write the expression for vector c.

c=OP+OQ

From Figure 1, write the expression for scalar s by the use of definition.

s=|OP||a|

Here,

|OP| is positive length of OP,

|a| is positive length of vector a.

From Figure 1, write the expression for scalar t by the use of definition.

t=|OQ||b|

Here,

|OQ| is positive length of OQ,

|b| is positive length of vector b.

From Figure 1, write the expression for vector c in terms of vectors a and b.

c=sa+tb (1)

The x and y-coordinates of vector c are estimated by the use of equation (1). Substitution of x-coordinated of vectors a and b along with scalar results x-coordinate of vector c and substitution of y-coordinates of vectors a and b along with scalar provides the value of y-coordinate of vector c.

Substitute a1 for a, b1 for b, and c1 for c in equation (1),

c1=sa1+tb1 (2)

Multiply a2 on both sides of equation (2).

c1a2=a2(sa1+tb1)

c1a2=sa1a2+tb1a2 (3)

Substitute a2 for a, b2 for b, and c2 for c in equation (1),

c2=sa2+tb2 (4)

Multiply a1 on both sides of equation (4).

a1c2=a1(sa2+tb2)

c2a1=sa1a2+tb2a1 (5)

Subtract equation (3) from (5)

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Chapter 12 Solutions

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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-40ESect-12.1 P-41ESect-12.1 P-42ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.2 P-1ESect-12.2 P-2ESect-12.2 P-3ESect-12.2 P-4ESect-12.2 P-5ESect-12.2 P-6ESect-12.2 P-7ESect-12.2 P-8ESect-12.2 P-9ESect-12.2 P-10ESect-12.2 P-11ESect-12.2 P-12ESect-12.2 P-13ESect-12.2 P-14ESect-12.2 P-15ESect-12.2 P-16ESect-12.2 P-17ESect-12.2 P-18ESect-12.2 P-19ESect-12.2 P-20ESect-12.2 P-21ESect-12.2 P-22ESect-12.2 P-23ESect-12.2 P-24ESect-12.2 P-25ESect-12.2 P-26ESect-12.2 P-27ESect-12.2 P-28ESect-12.2 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P-48ESect-12.6 P-49ESect-12.6 P-50ESect-12.6 P-51ESect-12.6 P-52ECh-12 P-1RCCCh-12 P-2RCCCh-12 P-3RCCCh-12 P-4RCCCh-12 P-5RCCCh-12 P-6RCCCh-12 P-7RCCCh-12 P-8RCCCh-12 P-9RCCCh-12 P-10RCCCh-12 P-11RCCCh-12 P-12RCCCh-12 P-13RCCCh-12 P-14RCCCh-12 P-15RCCCh-12 P-16RCCCh-12 P-17RCCCh-12 P-18RCCCh-12 P-19RCCCh-12 P-1RQCh-12 P-2RQCh-12 P-3RQCh-12 P-4RQCh-12 P-5RQCh-12 P-6RQCh-12 P-7RQCh-12 P-8RQCh-12 P-9RQCh-12 P-10RQCh-12 P-11RQCh-12 P-12RQCh-12 P-13RQCh-12 P-14RQCh-12 P-15RQCh-12 P-16RQCh-12 P-17RQCh-12 P-18RQCh-12 P-19RQCh-12 P-20RQCh-12 P-21RQCh-12 P-22RQCh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-1PCh-12 P-2PCh-12 P-5P

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